一道简单的数论问题

Question

Fill in each blank with an answer as easy as possible.
(a)M is a triangular number if and only if___________ is an odd square number.
(b)N is an odd square number if and only if____________is a triangular number.
triangular number:n(n+1)/2, square number:n^2.

Answer 1

(a)
M = n(n+1)/2
等价于：8M+1 = 4 n^2 + 4 n + 1 = (2n+1)^2
所以，第1个空应该填：8M+1

(b)
N = (2n+1)^2
等价于：(N-1) / 8 = n(n+1)/2
所以，第2个空应该填：(N-1)/8

Answer 2

8a+1
(b-1)/8